#1271372

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1759 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,6,5)(3,8,4,7)(9,10,14,16)(11,12,13,15)(18,21)(19,23)(27,30)(28,32);(1,3,6,4)(2,7,5,8)(9,11,14,13)(10,15,16,12)(25,26,29,31)(27,28,30,32);(9,12)(10,13)(11,16)(14,15)(17,18,20,21)(19,22,23,24)(25,27,29,30)(26,28,31,32);(1,4,6,3)(2,7,5,8)(9,12)(10,11)(13,16)(14,15)(25,28)(26,30)(27,31)(29,32);(1,5,6,2)(3,8,4,7)(9,13,14,11)(10,15,16,12)(17,19)(18,22)(20,23)(21,24)(25,28)(26,30)(27,31)(29,32)

Polycyclic presentation:

2^9|a^2=f,b^2=fhi,c^2=g,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1178568);